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Home » $ \frac137 \approx 0.027027 $ - Midis

$ \frac137 \approx 0.027027 $ - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 A lattice point \((x, y)\) satisfies both \(x, y \in \mathbb{Z}\) and \(xy = 100\). We seek integer solutions to this equation.

📌 Let \(d\) be a positive divisor of 100. Then \(x = d\) and \(y = \frac{100}{d}\) is an integer. Since \(xy = 100\), both \((d, 100/d)\) and \((-d, -100/d)\) are solutions.

📌 First, factor 100:

📌 \(100 = 2^2 \times 5^2\), so the number of positive divisors is \((2+1)(2+1) = 9\).

📌 The positive divisors are: \(1, 2, 4, 5, 10, 20, 25, 50, 100\).

📌 Each positive divisor \(d\) gives a solution \((d, 100/d)\), and each negative divisor \(-d\) gives \((-d, -100/d)\).

📌 Thus, total lattice points: \(2 \times 9 = 18\) (each positive divisor contributes two points: one in positive quadrant, one in negative).

📌 We verify: for each of the 9 positive \(d\), \((d, 100/d)\) and \((-d, -100/d)\) are distinct lattice points.

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